Related papers: A note on singularity categories and triangular ma…

Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced…

Let $\Lambda$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda$ admits a trivial maximal 1-orthogonal subcategory of $\mod\Lambda$, then for any indecomposable module $M \in \mod \Lambda$, we have that…

Given an ST-triple $(\mathcal{C},\mathcal{D},M)$ one can associate a co-$t$-structure on $\mathcal{C}$ and a $t$-structure on $\mathcal{D}$. It is shown that the discreteness of $\mathcal{C}$ with respect to the co-$t$-structure is…

We describe $\sigma$-matching, interchangeable and, as a consequence, totally compatible products on some classes of associative algebras, including unital algebras, the semigroup algebras of rectangular bands, algebras with enough…

We recognise Harada's generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with…

We give analogues of the Auslander correspondence for two classes of triangulated categories satisfying certain finiteness conditions. The first class is triangulated categories with additive generators and we consider their endomorphism…

We study some examples of braided categories and quasitriangular Hopf algebras and decide which of them is pseudosymmetric, respectively pseudotriangular. We show also that there exists a universal pseudosymmetric braided category.

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion…

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived…

Basic definitions and properties of nearly associative algebras are described. Nearly associative algebras are proved to be Lie-admissible algebras. Two-dimensional nearly associative algebras are classified, and its main classes are…

The aim of this paper is to construct singular equivalences between functor categories. As a special case, we show that there exists a singular equivalence arising from a cotilting module $T$, namely, the singularity category of $(^\perp…

We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type $\D$. They are $2$-CY-tilted algebras. Using a suitable process of mutations…

We introduce and study a category of algebras strongly connected with the structure of the Gelfand-Tsetlin subalgebras of the endomorphism algebras of Bott-Samelson bimodules. We develop a series of techniques that allow us to obtain…

We show that for an artin algebra $\Lambda$, the telescope conjecture for module categories is equivalent to certain idempotent ideals of mod-$\Lambda$ being generated by identity morphisms. As a consequence, we prove the conjecture for…

We study the relationship between singularity categories and relative singularity categories and discuss constructions of differential graded algebras of relative singularity categories. As consequences, we obtain structural results, which…

Let $A$ be an associative superalgebra over a field of characteristic zero. Let $n \geq d+1$. The main result of the paper establishes an equivalence of categories between supermodules for the wreath product $ S_{d} \wr A$ and an explicitly…

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a braided tensor category, we give a detailed construction of the…

A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual…

The Kuperberg Program asks to find presentations of planar algebras and use these presentations to prove results about their corresponding categories purely diagrammatically. This program has been completed for index less than 4 and is…

Let $\X$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $\ds(\underline\X)=\db(\mod\underline\X)/\kb(\proj(\mod\underline\X))$ of the stable category $\underline\X$ of $\X$. We…